A universal property of semigroup \(C^*\)-algebras generated by cones in groups of rationals

The article deals with the reduced semigroup \(C^*\) $C^{\ast }$ -algebras for the positive cones in ordered abelian groups. These \(C^*\) $C^{\ast }$ -algebras are generated by the regular isometric representations of the cones. Using the universal property of the isometric representations for the positive cones, we treat the reduced semigroup \(C^*\) $C^{\ast }$ -algebras as the universal \(C^*\) $C^{\ast }$ -algebras which are defined by sets of generators subject to relations. For arbitrary sequences of prime numbers, we consider the ordered groups of rational numbers determined by these sequences and the reduced semigroup \(C^*\) $C^{\ast }$ -algebras of the positive cones in these groups. It is shown that such an algebra can be characterized as a universal \(C^*\) $C^{\ast }$ -algebra generated by a countable set of isometries subject to polynomial relations associated with a sequence of prime numbers.